Problem: All of the 4th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$7.00$ each for teachers and $$3.50$ each for students, and the group paid $$56.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$14.00$ each for teachers and $$7.50$ each for students, and the group paid $$117.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7x+3.5y = 56}$ ${14x+7.5y = 117}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-14x-7y = -112}$ ${14x+7.5y = 117}$ Add the top and bottom equations together. $ 0.5y = 5 $ $ y = \dfrac{5}{0.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {7x+3.5y = 56}$ to find $x$ ${7x + 3.5}{(10)}{= 56}$ $7x+35 = 56$ $7x = 21$ $x = \dfrac{21}{7}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {14x+7.5y = 117}$ and get the same answer for $x$ ${14x + 7.5}{(10)}{= 117}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.